If the magnitudes of two vectos `vec(a) and vec(b)` are equal then which one of the following is correct?

Options

A. `(vec(a) +vec(b))` is parallel to `(vec(a) - vec(b))`
B. `(vec(a) + vec(b)) .(vec(a) -vec(b))= 0`
C. `(vec(a) +vec(b))` is perpendicular to `(vec(a)-vec(b))`
D. None of the above

Answer

To solve the problem, we need to analyze the relationship between two vectors \(\vec{A}\) and \(\vec{B}\) when their magnitudes are equal. Let's denote the magnitudes of the vectors as follows:

\[ |\vec{A}| = |\vec{B}| \]

### Step 1: Understanding the Magnitudes
Since the magnitudes of the two vectors are equal, we can express this mathematically as:

\[ |\vec{A}|^2 = |\vec{B}|^2 \]

### Step 2: Dot Product of the Vectors
Next, we will consider the dot product of the sum and the difference of these two vectors. We compute the dot product of \((\vec{A} + \vec{B})\) and \((\vec{A} - \vec{B})\):

\[ (\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B}) = \vec{A} \cdot \vec{A} - \vec{B} \cdot \vec{B} \]

### Step 3: Expanding the Dot Product
Now, we can expand the dot product:

\[ \vec{A} \cdot \vec{A} = |\vec{A}|^2 \quad \text{and} \quad \vec{B} \cdot \vec{B} = |\vec{B}|^2 \]

Substituting these into our equation gives us:

\[ |\vec{A}|^2 - |\vec{B}|^2 \]

### Step 4: Setting the Equation
Since we know that \(|\vec{A}|^2 = |\vec{B}|^2\), we can substitute this into our equation:

\[ |\vec{A}|^2 - |\vec{B}|^2 = 0 \]

### Step 5: Conclusion from the Dot Product
This implies that:

\[ (\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B}) = 0 \]

### Step 6: Interpretation of the Result
The result of the dot product being zero indicates that the vectors \((\vec{A} + \vec{B})\) and \((\vec{A} - \vec{B})\) are perpendicular to each other. This means that the angle \(\theta\) between them is:

\[ \theta = 90^\circ \quad \text{or} \quad \theta = \frac{\pi}{2} \]

### Final Answer
Thus, the correct conclusion is that if the magnitudes of two vectors \(\vec{A}\) and \(\vec{B}\) are equal, then the vector \((\vec{A} + \vec{B})\) is perpendicular to the vector \((\vec{A} - \vec{B})\).